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C3×C4×D4×3−1+2 #72254685

order2592
nilpotency class2
automorphism?
homologyAbelianization (H1) = C22×C33×C4 · Schur multiplier (H2) = C23×C32

Aliases

direct productC3×C4×D4×3−1+2

Subgroups

subgroupG/H
centerC2×C32×C4C22×C32
commutatorC2×C3C22×C33×C4
Frattini, cosocleC22×C3C23×C33
socleC22×C32C23×C32
prime radicalC2×C33×D4C2×C3
LCSG ▷ C2×C3 ▷ C1
UCSC1 ◁ C2×C32×C4 ◁ G

1786 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 432

A minimal faithful permutation representation — 3 generators on 20 points:

(1,2,4,3)(6,8)(13,15,18)(16,20,19);(1,3,4,2)(12,13,16,14,15,19,17,18,20);(5,6,7,8)(9,10,11)

Conjugacy classes & character table

660 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup ofC4×D4×C3⋊AGL(1,ℤ/9), #72254760
maximal subgroup ofC3×3−1+2×#442, C4×D4×C3⋊AGL(1,ℤ/9), #72254760
maximal normal subgroup ofC3×3−1+2×#442, C4×D4×C3⋊AGL(1,ℤ/9), #72254760
maximal quotient ofC3×3−1+2×#442

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2,4,3)(6,8)(13,15,18)(16,20,19), (1,3,4,2)(12,13,16,14,15,19,17,18,20), (5,6,7,8)(9,10,11));